UNITARY-MATRIX MODELS AS EXACTLY SOLVABLE STRING THEORIES

Models of unitary matrices are solved exactly in a double scaling limit, using orthogonal polynomials on a circle. Exact differential equations are found for the scaling functions of these models. For the simplest model (k=1), the Painlevé II equation with constant 0 is obtained. There are possible nonperturbative phase transitions in these models. The scaling function is of the form N-1/(2k+1)×f(N2k/(2k+1) for the kth multicritical point. The specific heat is f2, and is therefore manifestly positive. Equations are given for k=2 and 3, with a discussion of asymptotic behavior. © 1990 The American Physical Society.